Mode interaction in a spectrum of weakly unstable plasma waves

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Mode interaction in a spectrum of weakly unstable plasma waves

M. Trocheris

To cite this version:

M. Trocheris. Mode interaction in a spectrum of weakly unstable plasma waves. Journal de Physique,

1985, 46 (7), pp.1123-1135. �10.1051/jphys:019850046070112300�. �jpa-00210055�

Mode interaction in a spectrum ^of weakly ^unstable plasma ^waves

M. Trocheris

Association EURATOM-CEA

la fusion, Département de Recherches sur la Fusion Contrôlée, ^{Centre d’Etudes} Nucléaires, Boîte Postale n° 6, 92260 Fontenay-aux-Roses, ^France

(Reçu le 27 novembre 1984, accepté ^{le 5}

1985)

Résumé.

L’évolution d’un spectre d’ondes de plasma légèrement ^{instables en} présence ^{d’un faisceau} élargi

vitesse est traitée habituellement _{par la} théorie quasi ^linéaire. Cependant des simulations numériques ^{et des} expé-

riences de laboratoire ont révélé des écarts à la description quasi linéaire dans la présence ^de structures persistantes

dans l’espace ^des phases ^{et dans le} comportement individuel des modes. On montre dans cet article _que l’évolution inattendue des modes individuels peut ^être expliquée ^{à l’aide} d’une interaction des ondes par l’intermédiaire de

perturbations balistiques ^de grande longueur ^{d’onde. Le} point ^de départ ^est

théorie faiblement non linéaire qui comprend ^les parties balistiques ^des perturbations. ^{On établit d’abord}

système d’équations ^{de base} qui

foumit une généralisation de la théorie quasi ^{linéaire. En} simplifiant ^{le modèle} théorique ^{on arrive à un} système d’équations différentielles ordinaires qui ^{sont résolues} numériquement ^{dans des cas} représentatifs.

Abstract

The evolution of

spectrum ^of weakly ^unstable plasma ^waves in the presence of

beam is

usually ^treated by quasilinear theory. Numerical simulations and laboratory experiments, however, ^showed depar-

tures from the quasilinear description both in the presence of persistent structures in phase space and in the beha- viour of individual modes. It is shown in this paper that the unexpected ^evolution of individual modes can be

explained ^{in terms} of interaction of the waves via long

length ^free streaming perturbations. ^The starting point

is

weakly

^linear theory in which the free streaming portions ^{of the} perturbations ^{are included. A basic} system of equations ^{is first} derived which provides

generalization ^of quasilinear theory. The theoretical model is further

simplified ^{and leads to}

system ^of ordinary differential equations, ^which

^solved numerically ⁱⁿ representative

cases.

Classification Physics ^Abstracts

52. 35M

1. Introduction.

The weak warm-beam instability ^{in a} one-dimensional electron plasma has been the subject of extensive

theoretical, experimental and numerical work. Early

numerical simulations [1] already ^revealed departures

from the predictions ^of quasilinear theory [2], ^which

were confirmed by experimental ^studies [3] ^and by

recent numerical work [4]. ^The observations were

formulated and interpreted in several languages, ^but mainly ^{in terms of} unexpected field and particle-

motion correlations. The _presence of ordered struc- tures in phase space and of convective streams of

particles ^{was noted} [5] ^and compared ^{with a} theory

of moderate turbulence (see ^{Refs. [3] and [4] and}

references therein). ^{It was also observed and report-}

ed [4, 6] ^that single ^{modes do not in} general ^{show a}

smooth quasilinear ^{behaviour but that} they happen

to rise and decay ^rather abruptly. ^{The average evolu-}

tion of a group of neighbouring ^{modes was found,} however, ^to evolve much more regularly [6].

This body of numerical and experimental ^{data has}

lead to extensive theoretical work aimed at revising

and improving quasilinear theory. ^A thorough ^review

and a rigorous reformulation of quasilinear theory [7]

showed the importance ^{of mode} coupling and gave

a comprehensive statistical description of the evolu- tion of the spectrum of excited waves. An alternative

approach ^is considered in this paper, which is ^{not sta-} tistical in nature. The starting point ^{is a} weakly ^non

linear theory [8, 9], ^which applies ^to the evolution of small amplitude plasma ^{waves and whose} validity ^is

limited to a time scale of the order of the bouncing period ^of trapped particles in the field of the waves

considered. According ^{to this} theory, ^{the non linear}

evolution of the plasma ^{is described in terms of wave} amplitudes ^{and free} streaming perturbations. ^Non

linear interaction of free streaming perturbations ^with plasma ^{waves, once observed as the « linear side band}

effect » [11], ^has recently been demonstrated numeri-

cally ^and analytically [12]. ^{In the} present work, inter- action of neighbouring modes is found to occur via a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046070112300

long ^wave length ^free streaming perturbation, ^which

is created by ^the beating of the modes and interacti with them. This effect is reminiscent of ^non lineaj Landau damping, but it is distinct from it in two ways the interaction is due entirely to resonant particles ^anc

the important part ^is played by ^{the free} streaming por- tion of the beat disturbance rather than by its electric field. Non linear mode coupling ^was considered long

ago in the frame work of weakly ^non linear theories

[10], ^{but the free} streaming portions ^were consistently neglected, ^thus missing the main effect.

The physical ^situation chosen in this _paper is the

same as in reference [4]. ^{A one} dimensional electron

plasma ^of length ^L

1 024 À.D is considered and

periodic boundary conditions are applied. ^{The ions}

are replaced by ^{a uniform} neutralizing background.

The unperturbed distribution function fo(v) ^{has the}

form of a main Maxwellian plus ^{a warm beam :}

The values of the parameters ^{are also the same as in} reference [4] namely :

In numerical simulations, ^the unstable waves are most conveniently ^{allowed to} grow ^out of a small-

intensity ^initial noise, whereas in this _paper initial

amplitudes ^{of waves will be} given ^values correspond- ing ^{to the onset} of non linear effects. The values of the reduced amplitudes :

(where Ek’ ^{is the wave} electric field and no ^the unper- turbed density) ^will lie between 10- 3 and 10- 2 and

quasilinear ^{evolution will take} place for values of

wp t of a ^{few hundred}

The _paper is organized ^{as follows. In section 2 a}

basic system ^of equations ^is derived, ^{which is an}

extension of the usual quasilinear system ^of equations

in that it involves the free streaming perturbations ^of

small wave numbers in addition to the wave ampli-

tudes and to the _space averaged distribution function.

In section 3 this system ^of equations ^{is reduced to a} system ^of ordinary differential equations ^{on the wave} amplitudes, the values of the _space averaged ^distri-

bution function and the values of the free streaming perturbations ^{taken for the} phase velocities of the

waves. In section 4 the system ^of equations is modified

by applying ^a smoothing procedure ^{to the} velocity dependence ⁱⁿ analogy ^{to a} successful practice ⁱⁿ

numerical simulations [4]. ^The equation of the model

are integrated numerically and the results ^are _pre- sented and discussed in section 5.

2. Basic equations,

Let us briefly recall the results of the weakly ^{non linear} theory [8, 9] which is the starting point of this work.

A one dimensional electron plasma ^with period ^{L in}

space is considered and the distribution function

f(x, v, t) ^is expanded ^{in a} Fourier series in ^x in the usual _{way. In} order to treat the non linear interaction of plasma waves, it is convenient to be able to extract from each Fourier coefficient fk(v, t) ^the portion ^cor- responding ^to plasma ^{waves. This is} possible ^{in a} way which is suggested by the behaviour of fk(v, ^{t) in the}

linear theory ^of plasma waves, namely by ^its asymp- totic form for large ^{t :}

where A:, w: ^and uk denote the amplitude, frequency

and phase velocity ^{of the wave of wave number k cor-} responding ^{to root number a of the} dispersion ^rela-

tion. Ak ^and Pk(v) ^are independent ^{of t and are} easily expressed ^{in terms of} fk(v, 0). The full solution fk(v, t)

of the linearized equation ^{involves an additional} rapidly damped ^{term which can also be} expressed ⁱⁿ

terms of Pk(v). ^More generally ^{one can establish a} correspondence ^{between on the one hand} any suffi-

ciently regular ^function fk(v) ^{and on} the other hand a

set of amplitudes a: ^{and a function} t/Jk(V) ^{defined in}

such a way that they ^{have the above} simple ^time depen-

dence when computed ^{from the solution} fk(v, t) ⁱⁿ

the linear approximation. ^This correspondence ^can

then be used to make a change of unknown functions and translate the Vlasov equation ^{in terms of} ak(t) ^and t/lk(V, t).

The next step ^{in the} theory ^{is then to} apply ^an approximation ^{of the} Krylov-Bogoliubov type ^{to the} system ^of equations ^on all, qlk’ Writing

these quantities ^are of first order in the small _para- meter 8 of the theory and, ⁱⁿ addition, Ak ^and Pk ^are slowly varying functions of t. The small parameter is more precisely of the order of the ratio of the _per- turbed plasma density ^{to the} equilibrium density ^and

this turns out to be of the order of kÀ.D Êk ^or

H 2 where AD ^{is the} Debye length, E. ^{the reduced} amplitude (wB) ^of equation (3) and (oB the bounce frequency

of the trapped electrons in the wave potential. ^The

amplitude Ak ^{is related to the wave} potential 0’ ^and

The final equations ^{of the} weakly ^{non linear} theory

are equations (44.14) ^and (4.15) ^{of reference} [9] ^or equations (33) ^and (46) of reference [8]. ^{A detailed} analysis of their derivation shows that they ^{are valid}

over a time scale of the order of 1/((opl/1-C-) ^provided

contributions to Ak and t/Jk ^{of order} larger ^{than one in}

s are neglected. Additional approximations ^{will now}

be made in order to simplify ^these equations ^{for the} problem considered in this _paper.

Let us first consider equation (4.15) of reference [9]

which describes the creation of a free streaming per- turbation t/Jk(V, t) through the interaction of a wave of

amplitude ak, with another wave of amplitude ak" ^or

with a free streaming perturbation t/1k"(V, t). ^This equation ^{contains « secular terms »} growing propor- tional to t, which arise in the differentiation of 41k "(v, t)

with respect to v, when it is written in the form of

equation (4). ^The approximations ^to be made will consist essentially ⁱⁿ neglecting ^{terms which are small} compared ^to the secular terms. In so doing, ^{it is most} important ^{to make sure that the} neglected ^{terms are}

not singular for the values of the velocity considered,

in particular ^{for the} phase velocity ^u" of the k" wave.

To this end one can write equation (4.15) ^{of refe-}

rence [9] ^{in the} following ^form by discarding ^terms

that are both regular ^as functions of the velocity ^and

non secular in their time dependence :

where q and q" ^{are the} signs of k and k" and where the function Gk,,(v, ^{t) is} regular ^{in v and} slowly varying ⁱⁿ

time. It is seen that the term containing Gk" is less secular than the remaining part ^{of the} expression ⁱⁿ square brackets at least if v is close to u". Thus if a group of neighbouring ^{waves is excited and if v} lies in the _range of their phase velocities, the contribution of Gk" may be neglected. Further the values of k considered are small since

they ^are the differences of neighbouring ^{wave numbers,} and for small values of k, e* (k, v) ^are equal ^{to a} good approximation ^{so that the} factors 8" ^and En" ^{may be} dropped Writing ^the equation ^{in terms of the} slowly varying quantities Ak ^and Pk, ^we obtain the first basic set of equations :

with

and where co’ t and w" t have been written for brevity instead of the time integrals ^of w’(t) ^and w"(t).

The second set of equations, i.e., equation (4.14) of reference [9], gives the evolution of the wave amplitudes

under the influence of the free streaming perturbations. Neglecting ^{the mode} coupling terms, ^{which are far from} .resonant, this ^set of equations ^{reduces to :}

with

where Ck is the usual Landau path ^of integration.

The system ^of equations ^{must now be} completed by adding ^the equation for the space averaged ^distri-

bution function fo(v, t). ^This equation ^{can be} simplified

in the same way ^as the equation ^{for the free} streaming perturbation by keeping only ^{the most secular terms}

with a result similar to (7) :

where u stands for ulk’ and where values of P, y ^corres-

ponding ^to phase velocities Uk ^and uk ^{of the same} sign

should be retained.

The basic equations (7), (8), (10) apply ^{to the situa-}

tion considered where a spectrum ^of weakly ^unstable

waves is excited. This system ^of equations appears ^as

an extension of the classical quasilinear theory, ^{in that}

the time evolution of the wave amplitudes ^interacts

not only ^{with the} space averaged distribution function, but also with free streaming perturbations ^of long

wave length. The time scale over which this descrip-

tion is valid is essentially that of the underlying weakly

non linear theory, namely ^the bouncing period ^of

trapped electrons in a representative ^wave.

3. System ^of ordinary differential equations.

The basic equations ^{can be further} simplified by ^conti- nuing ^to select the most secular terms. Provided fo(v)

is not too small, ^the following approximation ^{can be}

made in equation (7) :

Then equation (7) ^{is no} longer differential in v,

which now enters merely ^{as a} parameter. As the deriva- tive ð/ðvPk is needed in equation (8), ^{one is lead to}

differentiate equation (7) ^with respect ^{to v and to make} the additional approximation :

Approximations ^{of the same} type ^{can be made on} equation (10) ^{and on this} equation differentiated with respect ^{to v. Thus a} complete system ^of equations ^is

obtained in which v enters as a parameter. According

to equation (8) ^the only relevant values of v are the

phase velocities of the excited waves, which we will consider to be a finite number corresponding ^{to the}

case of a plasma ^{of finite} length ^L.

The wave numbers are then multiples of k1 = 2 n/L

and the index k can be conveniently replaced by ^the integer n

k/k1. According ^{to the chosen form} (1) ^of fo(v), the excited waves correspond ^to positive phase

velocities and to a certain interval of I n I

The index, ^a, fl ^or y ^on the wave amplitudes ^and phase

velocities will be dropped ^{with the} understanding ^that only ^{waves of} positive phase velocities are considered

(then A-n

(An)*). ^{The free} streaming perturbations

are created with wave numbers ± mk1 with 1 , m

N2 - N1 ^{and it is} easily ^{seen that} only ^the P_ m ^enter equation (8). ^{The outcome of the} approximations ^is

then a system ^of ordinary differential equations ^{on the} following ^{unknown functions} of the time :

A few additional simplifications ^{can be made to this}

system ^of equations. First, ^if N2 - N1 N l’ ^{it is a}

consequence of the equations ^{that :}

and equation (8) ^{can be} simplified accordingly. ^Further

in differentiating equation (7) ^with respect to v, ^the

term involving 010v ^eikUt may be neglected.

Finally ^the following system ^of equations ^{is obtain-}

ed :

Finally ^the following system ^of equations is obtained :

t

with and the notations :

The frequencies ^{and the} phase velocities _{Cok and u,,,} ^are given by ^{the roots of the} dispersion relation and have

small imaginary parts :

In order to get ^{a closed} system ^of equations ^most simply, ^{one can take} Yn(t) proportional ^to fo(un, t) :

where Yn(O) ^{is the linear} growth ^rate corresponding ^to expression (1) ^for

It was assumed from the start that only ^free streaming perturbations of small » wave numbers ^came into

play, ^{which is the case if the} spectrum of excited waves is fairly ^{narrow so that} N2 - N 1 N 1. ^{This was used in} simplifying equation (6) ^and again ⁱⁿ simplifying equation (9) according ^to equation (14). ^This assumption ^is fairly ^well justified ^{in the} examples ^{to be discribed} in section 5, ^{but not} really ^{in the case} of the numerical simula- tion of reference [4], where the full spectrum of unstable waves is excited from the initial distribution function (1).

The system ^of equations (15-17, 22) ^{is still} meaningfull ^{in the case of a} single wave, when it reduces to a single equation ^on fo(un, t)

According ^{to this} equation, fo(un, t) ^would drop ^{to zero} very suddenly ^{after a} time of the order of OJB 1, ^denoting by COB the angular ^bounce frequency ^{of the} trapped electrons defined by equation (5). ^{It should be} noted, however,

that the approximations leading ^to equation (23) ^{are no} longer ^{valid once} fo(un, t) ^{has become too small. The}

same remark applies ^to equation (15-17) ^{whenever one of the} quantities fo’(u,,, t) ^{tends to zero.}

The case of two waves is also interesting. If n and n + 1 are the two wave numbers considered, equation (15)

and (16) ^{can be} brought ^{back to a second order} differential equation ^on An(t), ^{which takes the} following simple

form after some additional approximations :

A formal asymptotic expansion of the solution for large t shows that An(t) ^is likely ^to behave like :

with

which means that the relevant time scale for non linear interaction of the two waves is roughly WB 1.

4. Smoothing procedure.

In some numerical simulations it was found useful

to use a smoothing procedure ^{in order to avoid too} sharp dependences ^{on the} velocity leading ^{to numerical}

difficulties [4]. Typically ^the velocity distribution at every point in space would be smoothed over a small

velocity interval w (a small fraction of the thermal

velocity) ^after every time interval At. The smoothing procedure would be for instance :

and the results would _prove insensitive to the values of w and At within a suitable _range of values.

It was thought interesting ^to try and introduce ^an

analogous smoothing procedure ^{in the} analytical theory ^{to make the} comparison with numerical work

more realistic. In addition one may hope ^{that a reaso-}

nable smoothing procedure ^can suppress numerical difficulties in the solution of the differential equations

of section 3 without modifying the results substan-

tially. Such difficulties could probably be avoided by purely numerical methods in the code that solves the differential equations. ^{But it was} considered preferable

to incorporate ^a smoothing procedure ^{in the} analy-

tical theory ^{so as to} display ^its significance ^more clearly.

If there is _any need to smooth the velocity depen- dence, ^it certainly ^{arises in} expressions ^{of the form :}

that occur in the right ^{hand side of} equations (7) ^and (10) ^for Pk ^and fo, ^{with k", u" for k, u. These} expressions

are portions ^{of the} solution of the linearized Vlasov

equation ^{for wave} number k" and we will interpret fo(v) Hk(v, ^{t) as the} solution of the equation :

that vanishes for t

0 and we will examine the effect of a smoothing procedure of the form (27) ^{on this}

solution.

It is shown in Appendix A that this effect can be

simulated, ^{at least semi} quantitatively, by adding ^a

small diffusion term to equation (29) :

provided ^{Av and At} satisfy

An approximate solution of equation (30) ^{can be}

found for times limited by

and this is done in Appendix ^B. This solution is :

The analogue ^{of the} smoothing procedure ^will therefore consist in modifying ^{the functions} Hn(v, t)

and Hn(v, ^{t) defined} by equations (19) ^and (20) ^{in the} following ^{way :}

5. Numerical results.

The system ^of equations defined in sections 3 and 4 was

integrated numerically for various values of the _para- meters. In the absence of smoothing ^the equations

derived in section 3 can be integrated ^{with standard}

codes. Still the computation ^is usually stopped by

numerical difficulties which apparently ^{cannot be}

overcome by simply reducing ^{the time} step. ^{But as} will be seen below, whenever this happens ^the system of equations appears ^no longer ^{to be valid}

The unperturbed distribution function fo(v) ^was

chosen to have the form of equation (1) with the values of the parameters given by equation (2).

The initial amplitudes ^{of the} waves were given ^such

values that the interesting effects would occur on a

time scale for which the theoretical model is valid.

As was mentioned previously, this time scale is of the order of the bouncing period ² Tr/cos ^{of the} trapped

electrons in the waves considered. A suitable _range of values of the reduced amplitudes ^defined by equa- tion (3) ^then turns out to be between 10-3 and 10 - 2.

The calculations presented ^{in the} following ^{were made} with £

3 x 10-3 for all waves at t

0, ^{so that}

2 a/COB ²⁶⁰ cop ^1. With these initial amplitudes ^the

effects of non linear interaction are clearly ^observed

for wp ^t 100, ^{as well as} the evolution of the growth

rate.

The choice of the smoothing parameter, i.e., ^the

diffusion coefficient D in velocity space used in equa- tions (35, 36), ^{was also} inspired by ^{the numerical}

simulation of reference [4] ^{and most} of the calculations

were made for :

The values of ^D chosen in each particular ^{case will}

be defined with reference to this standard value.

5.1 THE TWO-WAVE PROBLEM.

In the case of two waves of wave numbers nk1 ^and (n ⁺ 1) k1, ^{the only} significant initial value is that of I Ên+ 1 I. The results

are ^{shown in} figure ^{1 for a} representative’case : n

^30,

Ên +1

^{3 x 10-3. The solid curves give} the evolution of E30(t) ^and E31(t) ^as obtained without smoothing procedure. ^The computing ^was stopped ^at wp t

⁵⁵

because the values of fo(un+ 1, ^{t) was becoming too}

small for the equations ^to remain valid. The dashed lines correspond ^to the calculation with standard

smoothing, ^{in which case the} computing ^was stopped

for the same reason at wp t

75. The behaviour observed with and without smoothing ^is qualitatively

the _same, although the difference is not negligible. ^The

decrease in amplitude ^{of the n}

^{30 wave is} clearly

seen and shows the importance ^{of the} type of interac- tion described by ^the system ^of equation of section 3.

This effect will become dramatic in ^cases with more

than two waves.

Fig. ^1.

^{Case of two}

^of

^numbers 30 k

^and

31 k18 The reduced amplitudes (defined by Eq. (3)) P3, ^and E31

^{plotted versus t} for initial values 3 x10-3 . The solid

curves show the results without smoothing ^{and the dashed}

curves

correspond ^{to standard} smoothing.

5.2 THE FOUR-WAVE PROBLEM.

With four waves

the situation is much more complicated ^{since free} streaming perturbations ^are created with three wave

numbers k1, ² kl, ³ k, and since the results depend sharply ^on the initial phases ^{of the waves.} The calcula- tions were made for four waves lying in the middle of the unstable spectrum ^{with n}

^{30 to} 33. The 33

wave is not affected by ^{the free} streaming perturbations

and the 30 wave on the contrary feels the action of the three wave numbers k1, ² k1, ³ k1. ^{In order to see the}

relative importance of the free streaming perturbations

of the three wave numbers calculations were made by truncating ^{the sum over m in} equation (15) to m ^M with M

1, 2, 3. The results are shown in figure ²

for a representative ^{case with} equal ^initial amplitudes

and phases of the four waves and with the standard

smoothing. ^{The solid curves} correspond ^{to M}

3, i.e., ^{to no} truncation. For the 32 wave the M = 2 and M

1 curves are not shown : in fact they ^are

very close to the M

3 curve shown, ^{as one would} expect since the 32 wave feels only ^{the effect of} the k1

free streaming perturbation. ^{For the 31 wave the two}

curves M

2 and M

3 almost coincide (only ^the

M

3 curve is shown), ^{but the M}

^{1 curve} (dotted line) ^is markedly different, ^{which means that the} 2 ki

free streaming perturbation ^{has a} large ^{effect on the}

31 wave. For the 30 wave the three curves M

1, 2, ³

are shown and the difference between M = 2 and

Fig. ^2.

^{Case of 4 waves of wave numbers} 30 ki ^{to 33} kl*

The reduced amplitudes t

plotted

^t for initial values of 3

10-3 and equal ^initial phases. ^Standard smoothing ^{is used Free} streaming perturbations ^of

numbers mkl (I m M)

^included. Dotted, ^dashed

and solid curves correspond respectively ^{to M}

1, 2, ^3.

M

3 is much less marked than between M

1 and M

2. Thus the effect of the 2 ki ^{wave number on the}

30 wave is large and the 3 ki ^wave number is less

important. ^A note-worthy ^{feature of} figure 2 is also the fact that all curves have a decreasing portion, starting

at Mp ^{t N} 30, ^so that the total energy of the ^waves decreases significantly ^before starting again ^{to increase} sharply.

A striking fact in the case of more than two waves

is the sharp dependence ^{of the} evolution of the ^waves

on their initial phases. Typical ^{results are shown in} figure ^{3 for the same excited waves} and initial ampli-

tudes as in figure 2, ^{but with two different sets of} phases.

Note in particular ^the very different behaviour of the 30 wave in figure ² (solid curve) ^{and in} figure ³ (dashed curve). Still another set of phases is used in figure ⁴

with the same initial amplitudes ^{and the} Ego ^{curve is}

seen to exibit a remarkable dip.

All curves shown in figures ^{2 and 3 were} computed

with the standard smoothing. ^{In order to} appreciate

the effect of smoothing, ^{results are shown in} figure ^{4 for}

the same initial data and with standard smoothing (solid curves), ^{half standard} smoothing (dashed curves)

and no smoothing (dotted curves). ^In the latter case the

Fig. ^3.

^{Four waves with the same}

^{numbers and}

initial amplitudes

ⁱⁿ figure ^{2. Solid curves} correspond

to the phases (0, 0, n, n) ^{and the dashed curves to the} phases

computation ^{was carried} only up ^to cop t

^{55 on}

account of too small values of fo(un). ^A fast decrease of fo(un) may be spurious and may be due ^to the break down of the approximations made in section 3, ^{so that} the results obtained with a certain degree ^{of smooth-} ing may be more representative of the solution of the

more exact equations ^of section 2. In order to docu-

ment this point, the evolution of fo(un) is shown in

figure ^{5 for n}

30 and 31 with the same parameters ^as in figure ⁴ and with the same meaning ^of solid, ^dashed

and dotted curves. The decrease is clearly ^faster

without smoothing, ^{but the curves} with standard and half standard smoothing ^{are similar. The curve}

n

31 with half standard smoothing ^{was carried} through up ^to wp t

¹⁰⁰ regardless ^{of the} excessively

small values of fo(un). ^{It can be} inferred from these results that the smoothing procedure, although ^{it has a}

non negligible effect, probably ^{leads to} qualitatively

correct results beyond the range of validity ^{of the} equations ^without smoothing.

5.3 THE MANY-WAVE PROBLEM.

A comprehensive description ^{of what} happens ^{with a} large ^number

of waves is impossible ^to give. ^But there remains the

Fig. ^4.

^{Four waves with the same wave} numbers and initial amplitudes ^{as in} figure 2 and with phases 0, 0, n/2 ’ 2 ’

Results with different values of the smoothing parameter

are shown : standard smoothing (solid curves), ^{half standard} (dashed curves) ^{and no} smoothing (dotted curves).

Fig. ^5.

^{Four waves with the same values of} parameters

as in figure 4. The evolution of fo(un) ^{is shown for n}

³⁰

and 31 and for the same values of the smoothing parameter

as in figure ^{4 : standard} (solid curves), half standard (dashed curves) ^{and no} smoothing (dotted curves).

general ^{fact that} very different behaviours of the waves can be observed by changing only their initial phases.

As was done in numerical simulations [4] ^the possible

evolutions can be explored by starting ^with randomly

chosen initial phases. ^{A number of runs were made}

with 8 waves and it appeared that anomalous beha- viour of a wave, i.e., ^a sharp dip ^{in the} Ên(t) ^{curve, was}

less frequent ^{than with 4 waves. A definite case of such}

behaviour is shown in figure 6, ^{where the curves} P.(t)

are shown for n

28 and 29 out of a set of 8 waves,

n

26 ^to 33. This figure ^{can be} compared ^{with a} typical result of numerical simulation (Ref. 4, Fig. 2)

and the behaviour of the two neighbouring ^{waves is}

seen to be strikingly ^{similar. It should be mentioned}

that the curves of figure ⁶ (solid lines) ^{were obtained}

with a fairly strong smoothing of four times the stan- dard. The dashed curves show the results with twice the standard smoothing, ^{in which case the} computa- tion could not be carried beyond wp t

^{60. In most}

cases with 8 waves, without _any markedly ^anomalous

behaviour of individual waves, ^no numerical diffi- culties were encountered with twice the standard

smoothing.

An attempt ^was also made at testing ^a statistical

prediction of reference [7] by running ^{a series of 8-wave}

cases with randomly ^chosen phases. The observed

growth ^{rate of each wave} yexp ^is predicted ^{to be} greater than the quasilinear growth ^rate ykl by ^{a certain}

factor provided ^suitable average values are taken for these growth ^{rates. For} yzxP ^the following weighted

Fig. ^6.

The evolution of two waves is shown out of

set of 8

(n

^{26 to} 33). ^{The solid curves} correspond ^to

4 times the standard smoothing and the dashed curves to twice the standard smoothing.

average is considered :

the sum extending ^{over the M cases} computed. ^The quasilinear growth ^{rate is} defined from the instanta-

neous space averaged distribution function fo(v, ^t)

also averaged ^{over the M} cases, which amounts to

taking ^the arithmetic mean of the M values of ykl(t)

for each k and ^t. Then the ratio yexp/yq’ should have a

definite value greater ^{than one,} which is estimated to be 2.2 in reference [13]. ^The validity of this result is

subject ^{to two} inequalities being satisfied, ^{the first}

of which is characteristic of a regime ^where quasilinear theory ^is invalidated by additional non linear effects [13, 14].

where Dkl ^{is the} quasilinear ^diffusion coefficient in

velocity space Dkl(v) ^{taken for v}

úJk/k. ^{The second} inequality ^{means that the waves are} weakly dispersive

and it can be written approximately [7,13,14] :

The values of QD, Yk and vd;sp ^{are shown in table I for a}

representative ^wave (n

29) and several values of the time. It is seen that the double inequality ^is marginally

satisfied in the middle of the time _range considered,

so that a qualitative ^test may be meaningful.

A series of 30 cases of 8 waves (n

^{26 to} 33) ^were

run with randomly ^chosen phases. ^{The other} para- meters were the same as for the cases considered pre-

viously ^{and a} smoothing of twice the standard value

was used. In three cases out of 30 the calculation was

stopped ^half way ^to the maximum time of 120 cop

and in three other cases it was stopped ^between

100 w; 1 ^{and 120} w; 1. The results on the growth ^rates

are shown in figures 7, 8, 9 ^{for a few waves in the middle}

Table I.

Fig. ^7.

^Time evolution of the quasilinear growth ^rate averaged

^{series of 30 cases of 8 waves with} randomly

chosen phases.

Fig. ^8.

^{Time evolution of the}

experimental

^average growth ^rate computed according ^to equation (38) ^{from the}

same cases

in figure ^7.

Fig. ^9.

^{Ratio of} the average growth ^{rates shown in} figures ^{7 and 8.}

of the spectrum. ^{First in} figure 7 ykl ^{is seen to decrease} gently ^{as one would} expect ^from quasilinear theory,

but not quite monotonously. ^The behaviour ^of yexp appears, however, ^{much more} complicated ⁱⁿ figure 8. The ratio y"PI-)Ak’ ^{is shown is} figure ^{9 and it}

does take values roughly equal ^{to those} predicted

for times greater than about 60 wp-1.

Acknowledgments.

The hospitality of the Centre de Physique Th6orique

de 1’Ecole Polytechnique during ^{the course of this}

work is gratefully acknowledged. The author also whishes to thank G. Laval and D. Pesme for stimulat-

ing discussions.

Appendix ^A

Let the smoothing operation ^on the function g(v, t) ^be performed ^at regular time intervals t1+ 1 - ti

^{At and}

assume this operation ^{to take the form :}

with

Then the procedure ^{is the} following ^when applied ^{to the} solution of equation (29) ^for g(v, t) :

If F(v) ^{is a} regular function of v which varies little over Av :

with

Thus :

and if ei’v " and fo(v) vary ^more slowly ^than g ^{with v :}

Now putting

and assuming ^{that terms of order} D(At)2 may be neglected, ^we get :

which is an approximate step by step ^solution of :

If W denotes the interval over which g ^varies appreciably ^{with v, the} conditions of validity of (A. 5) ^{are :}

Appendix ^B.

The solution of equation (30) ^for g(v, t) ^{can be found} by ^{means of a} Fourier transformation on v :

The transformed equation ^{is :}

and it can be solved by using ^u

^{y + kt as a new} variable. The solution that vanishes for t

0 is :

The solution for g is obtained as the inverse transform and can be written :

with

It turns out that K(v, t, s) ^{can be} easily computed ^if fo(v) ^{is a} Maxwellian :

and the result is :

with

It is seen that for

the expression ^for K(v, t, s) simplifies considerably ^{and reduces to} its value for D

0 :

so that the solution for g(v, t) ^{is :}

In fact this approximate ^solution for g is valid for ^more general ^functions fo(v) subject ^{to condition} (B. 5).

The same type of calculation can be made if fo(v) is for instance the nth derivative of a Maxwellian, ^still leading

to expression (B. 6) ^for g. Thus the approximate solution (B. 6) for g is valid when fo (v) ^is any linear combination

of Maxwellian functions multiplied by polynomials ^{of v.}

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